31-05-2012, 11:49 AM
Spanning Tree
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Suppose you have a connected undirected graph
Connected: every node is reachable from every other node
Undirected: edges do not have an associated direction
...then a spanning tree of the graph is a connected subgraph in which there are no cycles
Finding a spanning tree
To find a spanning tree of a graph,
pick an initial node and call it part of the spanning tree
do a search from the initial node:
each time you find a node that is not in the spanning tree, add to the spanning tree both the new node and the edge you followed to get to it
Minimizing costs
Suppose you want to supply a set of houses (say, in a new subdivision) with:
electric power
water
sewage lines
telephone lines
To keep costs down, you could connect these houses with a spanning tree (of, for example, power lines)
However, the houses are not all equal distances apart
To reduce costs even further, you could connect the houses with a minimum-cost spanning tree
Minimum-cost spanning trees
Suppose you have a connected undirected graph with a weight (or cost) associated with each edge
The cost of a spanning tree would be the sum of the costs of its edges
A minimum-cost spanning tree is a spanning tree that has the lowest cost
Kruskal’s algorithm
T = empty spanning tree;E = set of edges;N = number of nodes in graph;
while T has fewer than N - 1 edges {
remove an edge (v, w) of lowest cost from E
if adding (v, w) to T would create a cycle
then discard (v, w)
else add (v, w) to T
}
Finding an edge of lowest cost can be done just by sorting the edges
Efficient testing for a cycle requires a fairly complex algorithm (UNION-FIND) which we don’t cover in this course
Prim’s algorithm
T = a spanning tree containing a single node s;E = set of edges adjacent to s;while T does not contain all the nodes {
remove an edge (v, w) of lowest cost from E
if w is already in T then discard edge (v, w)
else {
add edge (v, w) and node w to T
add to E the edges adjacent to w
}
}
An edge of lowest cost can be found with a priority queue
Testing for a cycle is automatic
Mazes
Typically,
Every location in a maze is reachable from the starting location
There is only one path from start to finish
If the cells are “vertices” and the open doors between cells are “edges,” this describes a spanning tree
Since there is exactly one path between any pair of cells, any cells can be used as “start” and “finish”
This describes a spanning tree
Mazes as spanning trees
There is exactly one cycle-free path from any node to any other node
While not every maze is a spanning tree, most can be represented as such
The nodes are “places” within the maze
Building a maze I
This algorithm requires two sets of cells
the set of cells already in the spanning tree, IN
the set of cells adjacent to the cells in the spanning tree (but not in it themselves), FRONTIER
Start with all walls present
Pick any cell and put it into IN (red)
Building a maze II
Repeatedly do the following:
Remove any cell C from FRONTIER and put it in IN
Erase the wall between C and some adjacent cell in IN
Add to FRONTIER all the cells adjacent to C that aren’t in IN (or in FRONTIER already)